Anisotropic Ornstein noninequalities

Abstract

We investigate existence of a priori estimates for differential operators in L 1 norm: for anisotropic homogeneous differential operators T1, . . . , T ℓ , we study the conditions under which the inequalityholds true. We also discuss a similar problem for martingale transforms.

Remarks On Ornstein’s Non-Inequality In ℝ2×2

“…In some circumstances one can deduce the result for p = ∞ from the one for p = 1, see for instance [3,18] for the case P 2 = div, and in fact the result for p = 1 is much more difficult. Similar results also hold in the anisotropic setting, see [13,23].…”

Existence of minimisers of variational problems posed in spaces of mixed smoothness

“…Moreover, all the maximal elements of the smoothness lie on a common hyperplane {α : α, a −1 = 1}, thus allowing for convenient scaling, which we will discuss later. This hyperplane is called a pattern of homogeneity by Kazaniecki, Stolyarov, and Wojciechowski in [45]. It is worth mentioning that their paper is the first one to introduce a simple version of anisotropic quasiconvexity (to be discussed later) thus inspiring the present work.…”

“…If, on the other hand, the parities of |α| do not match, then the coefficients in (6.6) complexify, and we do not get any directional convexity. It is still possible to use the calculation leading to determining the form of the vectors in (6.6) to show that a-quasiconvex functions are (pluri)subharmonic in a certain sense (see the discussion in [45]), but we have not yet been able to use that to strengthen our relaxation results. We believe, however, that this should be studied further and intend to do so in future work.…”

“…We have already mentioned that, in general, there is no good analogue of rank-one convexity in the mixed smoothness setting. However, if a is such that all multi-indices α with α, a −1 = 1 are of the same parity, this has been resolved by Kazaniecki, Stolyarov, and Wojciechowski in [45]. Under this assumption they have shown that a-quasiconvex functions are convex along directions of the form…”

The Regularity of Minima for the Dirichlet Problem on BD

“…For a given open set ⊂ R n , this space consists of all u ∈ L 1 ( ; R n ) such that the distributional symmetric gradient ε(u) := 1 2 (Du + Du ) is a finite, matrix-valued Radon measure on . By Ornstein's Non-Inequality [24,50,51,65], there exists no constant c > 0 such that Dϕ L 1 ( ;R n×n ) c ε(ϕ) L 1 ( ;R n×n ) holds for all ϕ ∈ C ∞ c ( ; R n ). In consequence, BD( ) is in fact larger than BV( ; R n ), and the full distributional gradients of BD-maps in general do not need to exist as (locally) finite R n×n -valued Radon measures.…”